108 M. HUTCHINGS AND C. H. TAUBES, SEIBERG-WITTEN EQUATIONSIt is a consequence of the definition that the fiber Ex of E at any x E X (that
is, 7r-^1 ( x)) has, in a natural way, a vector space structure of e n. This is why E is
called a vector bundle.
In particular, it follows from the definition that the space of sections of E forms
a vector space, where (s 1 +s2)(x) = s 1 (x)+s2(x), and where >-·s1(x) = m.>,(s1(x))
for>-EC. (We sometimes use m.>, to denote the action of>-on E.) In this regard,
note that there is a canonical section, the "zero section: 0 : X --+ E which is a
bijection onto the image of multiplication by 0 on E. Note that ifs is a section and
f : X --+ e a smooth function, then f · s is a section.
One can make new vector bundles out of old ones by analogy with the standard
operations on vector spaces. For example if E, F are vector bundles over X , then
we can define Hom(E, F) to be the set of (x , h) such that x EX and h: E x --+ Fx
commutes with thee action. Likewise, the dual bundle E* consisting of (x, h) such
that h : E x --+ C and h commutes with the e action. We also have the tensor
product E 0 F = Hom(E, F).
A fundamental example of a (real) vector bundle is the cotangent bundle
T X --+ X. We let Dk(X) = C^00 (/\kT* X) denote the space of differential forms
on X. If Eis another vector bundle on X, we can consider "E-valued differential
forms", namely Dk(X, E) = C^00 (/\kT* X 0 E).
One more basic construction is the following.Definition 1.2. If 7r : E --+ X is a vector bundle and f : Y --+ X is a map, the
pullback bundle J* E--+ Y is the fiber product of 7r and f,
J* E _I. E
1 1 ~
y __.!___. x ,
i.e. j E = {(y, e) I y E Y, e E Ef(y)}, and the projection J E --+ Y sends
(y, e) I-+ y.There is a natural map j : J* E --+ E defined by
f (y, e) = e.
For each y E Y , J restricts to an isomorphism (J* E)y--+ Ef(y)·
1.2. ConnectionsIn general, there is no canonical isomorphism of a fiber of a vector bundle with en.
Let us explore what this means at the infinitesimal level, i.e. at the level of tangent
bundles. If 7r : E--+ Xis a vector bundle, there is an exact sequence of (real) vector
bundles on the space E ,
( 1.1) 0 ----> 7r E _:, TE ~ 7r TX ----> 0.
In this sequence, 7r E is the bundle whose fiber over e E E is the vector space in
which e lives, namely E~(e)· There is a natural map i from 7r E to the tangent
bundle TE, whose image consists of the "vertical" tangent vectors in E. (The map
i sends v E E to the tangent vector ft I t=O ( e + tv).) And 7r * is the derivative of 7r.
This sequence does not have a canonical splitting. In other words, there is no
canonical way to choose a "horizontal" subspace of TE complementary to the ver-
tical subspace 7r* E. However we can choose a splitting; and if the splitting satisfies